A157978 -id:A157978 - cp's OEIS frontend

A106015 Primes p such that 4*p +- 3 are primes.

2, 5, 11, 19, 59, 89, 109, 149, 151, 331, 359, 389, 401, 439, 499, 521, 571, 599, 829, 941, 1019, 1039, 1129, 1249, 1279, 1319, 1381, 1451, 1669, 1741, 1871, 2131, 2161, 2179, 2251, 2459, 2819, 3119, 3251, 3469, 3539, 3581, 3659, 3911, 4001, 4231, 4261

Offset: 1

Zak Seidov, May 05 2005

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

[p: p in PrimesUpTo(100000)| IsPrime(4*p-3) and IsPrime(4*p+3)]; // Vincenzo Librandi, Nov 13 2010

Select[Prime[Range[400]], PrimeQ[4#+3]&&PrimeQ[4#-3]&]
Select[Prime[Range[600]],AllTrue[4#+{3,-3},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 18 2021 *)

A023213 INTERSECT A157978. - R. J. Mathar, Jul 25 2009

A163182 Primes p such that neither 4p+3 nor 4p-3 are prime.

Original entry on oeis.org

3, 13, 43, 53, 73, 83, 97, 127, 137, 139, 163, 167, 173, 197, 199, 211, 223, 251, 269, 277, 281, 293, 311, 317, 337, 347, 379, 383, 397, 409, 419, 421, 433, 443, 449, 463, 491, 503, 547, 557, 563, 593, 601, 607, 613, 617, 641, 643, 727, 733, 757, 787, 809

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 22 2009

Primes neither in A023213 nor in A157978.

			For p=3, 4*3+3=15 (not prime) and 4*3-3=9 (not prime), so the prime p=3 is in the sequence.
For p=7, 4*7+3=31 (prime), so the prime p=7 is not in the sequence.

Marius A. Burtea, Table of n, a(n) for n = 1..5673

Cf. A023213, A106015, A157978.

lst={};Do[p=Prime[n];If[ !PrimeQ[2*p+(p-1)+(p-2)]&&!PrimeQ[2*p+(p+1)+(p+2)], AppendTo[lst,p]],{n,3*5!}];lst
Select[Prime[Range[150]],NoneTrue[4#+{3,-3},PrimeQ]&] (* Harvey P. Dale, Aug 01 2022 *)

isok(p) = isprime(p) && !isprime(4*p+3) && !isprime(4*p-3); \\ Michel Marcus, Oct 12 2018

Edited by R. J. Mathar, Jul 25 2009, Jul 27 2009

A306616 Integers k such that phi(Catalan(k+1)) = 4*phi(Catalan(k)) where phi is A000010 and Catalan is A000108.

Original entry on oeis.org

2, 8, 19, 20, 36, 42, 44, 55, 56, 76, 91, 109, 116, 120, 140, 143, 152, 156, 176, 184, 200, 204, 213, 216, 224, 235, 242, 260, 289, 296, 300, 380, 384, 400, 401, 415, 436, 464, 469, 476, 524, 547, 553, 564, 595, 602, 616, 624, 630, 631, 660, 685, 704, 716, 744, 776, 800

Offset: 1

Michel Marcus, Mar 01 2019

nonn

Integers k such that A062624(k+1) = 4*A062624(k).
Consists of integers k (see p. 1405 of Luca link):
k = 2p-2, where p >= 5 is a prime such that q = 4p-3 is also prime (see A157978);
k = 3p-2, where p > 5 is a prime such that q = 2p-1 is also prime (see A005382).

			phi(C(2)) = phi(2) = 1 and phi(C(3)) = phi(5) = 4 so 2 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Florian Luca and Pantelimon Stanica, On the Euler function of the Catalan numbers, Journal of Number Theory, Vol. 132, No. 7 (2012), pp. 1404-1424.

Cf. A000010, A000108, A062624.

Cf. A005382, A157978.

Select[Range[1000], EulerPhi[CatalanNumber[#+1]]== 4*EulerPhi[CatalanNumber[#]] &] (* G. C. Greubel, Mar 02 2019 *)

C(n) = binomial(2*n,n)/(n+1);
isok(n) = eulerphi(C(n+1)) == 4*eulerphi(C(n));

[n for n in (1..1000) if euler_phi(catalan_number(n+1)) == 4*euler_phi(catalan_number(n))] # G. C. Greubel, Mar 02 2019

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A106015 Primes p such that 4*p +- 3 are primes.

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A163182 Primes p such that neither 4p+3 nor 4p-3 are prime.

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A306616 Integers k such that phi(Catalan(k+1)) = 4*phi(Catalan(k)) where phi is A000010 and Catalan is A000108.

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